Quantitative Landis-type result for Dirac operators

We study quantitative unique continuation at infinity for Dirac equations with bounded matrix-valued potentials. For the massless Dirac operator ${\mathcal{D}}_n$ in ${\mathbb{R}}^n$, we establish a Landis-type estimate showing that the vanishing order of any nontrivial bounded solution of $({\mathcal{D}}_n+\mathbb{V})\phi =0$ satisfies a lower bound of order $\exp (-\kappa R^2(\log R{)}^2)$ as $\mathrm{\mid }x\mathrm{\mid }=R\to \mathrm{\infty }$; the quadratic growth in the exponent is sharp, in view of previous known results. Our proof follows a Bourgain–Kenig type approach based on a Carleman inequality for Dirac operators which relies on a local Hölder regularity result, which we also prove. In two dimension, we obtain improved quantitative estimates under symmetry assumptions on the potential $\mathbb{V}$ and for real-valued solutions. Finally, we also derive qualitative Landis-type results for Dirac equations with decaying potentials, including critical decay rates.